Yes, they can ticket you for this without doing a breathalizer test or searching everyone. The mere presence of alcohol among everyone at the party is enough for a citation of underage possession of alcohol for everyone at the party. Since your ticket does not state anything about intoxication, it is irrelevant whether you were drunk or not, you were not cited for any sort of intoxication.
Be advised: A conviction for possession of alcohol by a minor will result in a license suspension and increased car insurance rates even though you were not in a car. You need to contact a lawyer to handle this.